A relation $\displaystyle \mathrm{R}$ is defined on a set of real numbers $\displaystyle \mathbb{\mathrm{R}}$ as \[\mathrm{R}=\{(\mathrm{x}, \mathrm{y}): \mathrm{x} \cdot \mathrm{y} \text { is an irrational number }\} . \] Check whether $\displaystyle \mathrm{R}$ is reflexive, symmetric and transitive or not.

For reflexive $\displaystyle (1,1) \notin \mathrm{R}$ as $\displaystyle 1^{2}$ is rational R is not reflexive For symmetric Let $\displaystyle (\mathrm{x}, \mathrm{y}) \in \mathrm{R} \therefore \mathrm{x} . \mathrm{y}$ is an irrational number ∴ ( $\displaystyle \mathrm{y} . \mathrm{x}$ ) is an irrational number $\displaystyle \therefore(\mathrm{y}, \mathrm{x}) \in \mathrm{R}$ $\displaystyle \therefore \mathrm{R}$ is symmetric For Transitive $\displaystyle (1, \sqrt{2}) \in \mathrm{R},(\sqrt{2}, 2) \in \mathrm{R} \quad$ but $\displaystyle (1,2) \notin \mathrm{R}$ $\displaystyle \therefore \mathrm{R}$ is not transitive